Finite-Frequency Fluctuation-Response Inequality

We derive an inequality relating the finite-frequency linear response and fluctuations of an observable in a physical system in steady state. The relation holds for arbitrary observables and perturbations in general Markovian dynamics, including overdamped and underdamped Langevin systems and jump processes, both in and out of equilibrium. As a consequence, we obtain a universal upper bound on the broadband signal-to-noise ratio of noisy dynamics, which only depends on the damping constant and temperature. We further show that the inequality reduces to an equality for appropriately chosen observables or perturbations in linear systems, both overdamped and underdamped and both in and out of equilibrium.